The actuarial problem, by that I am assuming you mean solvency, is caused by borrowing from the trust fund and spending it in the general budget.
Your analysis is deeply flawed. It is exactly the reverse of what you said. Social security taxes actually result in the govt borrowing at 0.86% because that is the real return on social security. No other sector of the economy can borrow at that rate. The (real) equity premium over 30-40 years during any period from 1926-now is much higher than that.
Also, the benefit of social security privatization is like getting a tax-cut on your labor income. If you know that a marginal hour of labor will produce benefit that can be put in a private account that is for yours to keep, then people will work longer. The marginal investment in personal accounts from working a marginal hour is not a tax, as it is the case now, but a tax-cut.
As I said, it will bring about people chaning their labor input and also will make people more risk-averse and people will engage in risky activities less and less.
Ok.
If you know that a marginal hour of labor will produce benefit that can be put in a private account that is for yours to keep, then people will work longer.
Why would they work longer? I don't see anything in your reasoning to support this hypothetical syllogism. I mean, your conclusion does not follow from the premise. Some people might well prefer to go fishing. It only takes one to spoil the argument.
As I said, it will bring about people chaning their labor input and also will make people more risk-averse and people will engage in risky activities less and less.
Again, this requires quite a leap. Will a twenty-two year old high school graduate be more or less likely to be a risk taker than a forty-five year old high school graduate?
This seems to assume a level of rationality I simply do not believe we commonly share.
The government borrows at rates higher than that. The return on SS is simply a function of the generousity of the benefits. It is a residual number. It is generated by demographics and the claim for benefits. It has nothing to do with rates of return on investment. The equity premia of the past was generated by much higher dividend rates than now, and a doubling of the PE ratios. It won't be that high in the future. It will be closer to 3% rather than 6%. See ya.